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I know that the PoincarĂ© conjecture was first proved in dimension ≥ 5, then dimension 4, and finally 3. I'm just curious, does the Ricci flow approach by Perelman shed any light on the high dimension case? Or for some reason one cannot hope for a uniform proof because these dimensions are essentially different?

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Note: if the Ricci flow on a homotopy sphere did something like you said, it almost certainly would give you that your manifold was diffeomorphic to the sphere, not just homeomorphic. Morally, this is because (perhaps after surgeries) the flow should give you a positive constant curvature metric on the manifold, which is then diffeomorphic to the sphere. But this statement is still open in dimension $4$, and plain false in general dimensions bigger than or equal to $7$. – YangMills Oct 26 at 13:47
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 One may think of Ricci flow with surgeries as an algorithm. Since there is no algorithm to say that more-than-three-dimensional manifold is simply connected, you do not have much chance. (It will not help even if an old man told you that manifold is simply connected and you trust him.) – Anton Petrunin Oct 26 at 15:43
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 An other reason, is that in dimensions 5 and up, the neck surgeries do not lead to a simpler manifold. At least in principle, they can cancel each other. – Anton Petrunin Oct 26 at 15:44
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 There's Brendle-Schoen: ams.org/journals/bull/2011-48-01/… You could also consider Ricci flow on cohomogeneity one metrics, which exist on Milnor spheres: arxiv.org/abs/math/0601765 – Agol Oct 26 at 15:45
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 This is in the opposite direction to the original question, but the Ricci approach also works in dimension 2: arxiv.org/abs/math/0505163 . (And in dimension 1 as well, I suppose, though this is a vacuous truth.) – Terry Tao Oct 26 at 17:43
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